Like this:

Keith Devlin has posted yet another blog post on how multiplication is defined. I left a comment there, but it wasn’t as good as I hoped it would be. Now that I’ve gone back and read all the comments, I am completely convinced that I know exactly where the misunderstanding is occurring. I should maybe throw in that the comments have been turned off for that post. Maybe my blog doesn’t get enough traffic, but I can’t ever, ever see a reason to do that. You filter out a few comments that are off topic, but don’t shut down comments completely. It seems to go in blatant opposition to the idea of using a blog rather than just publishing somewhere. But that is a completely side issue.

Devlin wants to claim that the definition of multiplication is to define a function in Peano arithmetic using recursion. Since the “definition” as repeated addition does not yield a function on the whole natural numbers it cannot possibly be the definition. In this strictest sense, the definition is not repeated addition because it is this other recursively defined function.

I’m with him on all of this. Of course that is true. Repeated addition only works for a finite stage and you must relying on something else to get the full function. The place we disagree is in what we mean by “definition”. Devlin wants to say that because it is necessarily the case that to define an honest multiplication function axiomatically in Peano arithmetic we must make some part of the definition that is not repeated addition, then we must say that the definition is not repeated addition.

That is a completely absurd idea. Devlin is conflating the two mathematical notions of definition and existence. You can make a definition of anything you want in math, but that doesn’t mean it exists. You then should prove that such a thing exists. I claim that the above “not repeated addition” definition of a function is actually the proof of the existence of a multiplication function and not a definition.

Recall that the context of this conversation is all about whether or not we should teach children that multiplication is repeated addition. The whole point of teaching children multiplication is precisely so that they can apply it to real world situations. The only sense in which multiplication can be applied to real world situations is if we know that it produces the same answer as repeated addition. I talk more about that in the original blog post. Even these strange analogies about stretching being a continuous rather than a discrete concept only makes sense after interpreting it as repeated addition.

Now that we’ve established the difference between definition and existence I’ll decompose what is going on to show that multiplication is repeated addition (by definition and hence ought to be taught to children because it isn’t incorrect). Recall that definitions in mathematics are technically biconditional statements. Definition: A function is multiplication if and only if it satisfies repeated n times for any choice of in . Note that this is not only the definition that will tell us the multiplication function is useful, but a priori we don’t know that such a function exists.

Devlin’s post uses recursion to prove that such a function exists, but once these two notions are pulled apart we see that Devlin is actually incorrect. The (useful) definition is the one I give above. If we try to conflate the existence and definition as Devlin does, then we have no idea if the recursively defined can be used to compute something in reality. As I point out in the comments at that blog post, for all we know that nice recursive definition could yield for all inputs. Are we going to call that multiplication? I think Devlin wrote that off as a flippant comment to be ignored, but I still think it illustrates the necessity of separating definition from existence perfectly and is precisely our point of disagreement.

Now to try to summarize briefly, it seems that when Devlin wants to use the phrase “multiplication is not defined to be repeated addition” he seems to mean that in order to have a function defined on it must be defined recursively without reference to repeated addition. I agree completely. I just disagree that the definition of such a function should be consider the definition of multiplication (is the confusion that “definition” is used in reference to “defining a function” versus defining the term “multiplication”?). The actual definition should be the one I gave (otherwise we’ve made a useless function), and then the proof that such a function exists is what Devlin wants. I don’t see how Devlin could disagree with what I’ve written here, and hence I see no way to conclude anything other than multiplication is repeated addition…unless we are going to use some non-standard idea of what it means to “be” something.

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